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So welcome back everybody. Last time we introduced propositional and predicate logic as our language

with the aim to formulate set theory and that's precisely what we're going today.

And first we have a short chapter on the epsilon relation and in fact set theory as an axiomatic theory

is formulated on a fundamental postulate

namely on the postulate that there is a fundamental relation epsilon

that there is a fundamental relation.

Now what's a relation? A relation is just a short term for a predicate of two variables.

That is a predicate of two variables.

It's just quicker to say relation.

It's that there is a fundamental relation called epsilon.

Well obviously this will be the element symbol that we all expect to make its appearance in set theory

but let's not have any preconceived notions about it.

It's simply a relation and it's called epsilon.

There will be no definition of what epsilon is and that's important.

There will be no definition in the stricter sense of what epsilon is apart from it being some predicate of two variables

or for that matter or of what a set is.

Now this is weird. We're going to write down set theory but I tell you there will be no definition of what the elementary symbol epsilon,

the element symbol actually is, or what a set is.

Well I use the definition in a strict sense namely saying epsilon is defined like da da da da or a set is da da da da da.

Instead we're going to say this. Instead we're going to write down nine axioms.

Instead nine axioms that use or that speak of epsilon and sets.

And these eight axioms will teach us how to use epsilon and what constitutes a set.

So it will only be the interplay between the epsilon and what we call sets

which defines what a set is and which defines what the epsilon means.

And as I indicated last time such an approach is necessary if we want to start writing down mathematics from scratch

without any prior notions, without any prior terms in terms of which we could define the element relation or a set.

Now which axioms are there? So overview over the axioms.

So I memorize them as ee perp, ee perp, icf.

So you've got to learn this like a poem. ee perp, icf.

Well these ee, let's call them basic existence axioms. So that's their character.

And that has to do with the kind of a property of the element relation and the e stands for existence of an empty set.

So of course we'll specify this. The perp, they are heavily used construction axioms.

Construction axioms which instruct us how to build new sets once we have some given sets.

That's of course a recurring theme that if you've got a structure you want to know how to build from elementary building blocks

given structures of this type, new structures.

P is the pair set axiom, u is the union set axiom, r is the replacement axiom, powerful thing.

P is the power set axiom.

Now I see these are further existence axioms, further existence axioms, but they could also be classified as construction axioms.

That are a little more advanced and actually are rather newer than the other axioms.

And the f is the so-called axiom of foundation and the character of this is more of a non-existence axiom that says don't use sets of the following kind.

So that excludes something. That's the overview of axioms.

And now we're discussing them in turn and I will discuss them in this order partly because some of the later axioms rely on the earlier axioms.

But then also I want to give some immediate application of each axiom so that we get a feeling for them.

And sometimes I push them down the list in this order in order to be able to use some previous application in the more advanced application.

So that's why we have this order, but ee perp icf you agree is a wonderful mnemonic.

So never forget nine of them. These are the initial letters if you will.

So well before we go there let's actually introduce some new relations using the element relation.

I sometimes call it the element relation.

The element relation we can immediately define.